If f(x)= 0 has two roots, then, by Rolle's theorem, there exist a value of x between those roots at which f' is 0. Quoting from the Wikipedia article mr. fantastic linked to
"4. It is clear from the formula for Newton's method that it will fail in cases where the derivative is zero. Similarly, when the derivative is close to zero, the tangent line is nearly horizontal and hence may "shoot" wildly past the desired root."